Can I make clothes be weight-painted and have it have a cloth-simulation?
You have several solutions to animate a skirt: This one that uses Mesh Deform modifier. Or you can try with the Wiggle Bones addon. But if you want to use physics, use Cloth:

Create your skirt, it needs to be another object that the body, parent it to the first spine bone, subdivide it a bit, weight paint the top edge in order to create the pin group:



Give it a Physics &gt; Cloth simulation, in Shape &gt; Pin Group, choose the group you've created:



In the modifiers stack, put a Subdivision Surface modifier under the Cloth modifier. Select the body and give it a Physics &gt; Collision. Move, the armature, it should work: